Revisiting Differential Geometry (Pt. 4): Parametric Angle Variables for Surfaces

 To deal with surfaces and curves on them we start with a real, single-valued vector function and parametric angle variables using superscripts such that we have:  u¹ ,  u ² , whence:

(1)               x ( u¹ ,  u ² )  =

{ x ₁  ( u¹ ,  u ² ),  x ₂ ( u¹ ,  u ² ) , x ₃ ( u¹ ,  u ² ) }

With the two angular variables  u¹ ,  u ²  defined in a bounded domain B of the u¹  u ² -plane   Then by equation (1) to any point ( u¹ ,  u ² )   in B  there is associated in  R ₃  a  point  with position vector x (u¹ ,  u ²). Then the point set in  R ₃  obtained when u¹ ,  u ²  vary is called a parametric representation of the set and u¹ ,  u ²   are the parameters of this representation.

In these descriptions we will also make use of the Jacobian matrix of rank 2:

 J =

(¶ x ₁ /¶ u¹     ¶ x ₁ /¶ u ²  ) 

(¶ x ₂ /¶ u¹     ¶ x ₂ /¶ u ² ) 

(¶ x ₃ /¶ u¹     ¶ x ₃ /¶ u ² ) 

Partial derivatives of x ( u¹ ,  u ³ )  can then be denoted:

x ₁   = ¶  x/ ¶ u¹      And: x ₁₃   = ¶ ² x/ ¶ u¹  ¶ u³          

Which can also be more generally denoted:

x _a  = ¶  x/ ¶ u¹      

 And:

x _ab = ¶ ² x/ ¶ u_a¹  ¶ u _b³          

Or in the 2^nd case:

 x _ab = ¶ ² x/ ¶ u ^a¶ u ^b          

Note there are three determinants of 2^nd order in the Jacobian J which bear components of the vector products , i.e. of the vectors:

x ₁   = ¶  x/ ¶ u¹        and  x ₂   = ¶ x/ ¶ u²        

These are different from the null vector IFF  x ₁  and x ₂   are linearly independent vectors . 

The assumption that the matrix for Jacobian J is of rank 2 is thus the necessary and sufficient condition that said vectors are linearly independent.

Note the x ₁ x ₂ - plane can be represented in one of two forms, Cartesian:  

(i)                x ( u¹ ,  u ² ) =   (u¹ ,  u ²,  0),   

or polar

(ii)              x ( u¹ ,  u ² ) = ( u¹ cos u ² , u¹ sin u ², 0), 

For which the respective matrices are:

(i) J =  

(1   ....0)

(0 ......1)

(0......0)

And (ii), 

 J =

( cos u ² , -u¹  sin u ² )

( sin u ² ,  u¹  cos u ² )

(0 ....................    0  )

Recall the sphere can be obtained by extending the circle by one dimension ( x ₂ ), i.e. to obtain:

x ₃  = + Ö( r ²   -   x ₁ ² -   x ₂ ²)

Depending on sign this is a representation of one of two hemispheres,  x ₃  >  0  or   x ₃  <   o. The parametric angular representation of the preceding can be expressed:

x ( u¹ ,  u ² )  =  (r cos u ²  cos u¹, r sin u ² sin u¹ , r sin  u ²)

or writing out in terms of  x ₁ ,  x ₂ , x ₃ .

x ₁ = r cos u ²  cos u¹,  x ₂  =  r sin u ² sin u¹ 

x ₃ = r sin  u ²

The geometrical configuration is shown below:

    The similarity to the celestial sphere of astronomy will not be lost on many readers, i.e. the angular parameter u¹  plays an analogous role to Right Ascension while  u ²  plays a role analogous to the zenith distance (z) which is related to the coordinate of declination ( d  = 90 - z). Via this analogy we can also see the corresponding Jacobian matrix at the poles is:

J =

(-r cos u ²  sin u¹        -r sin u ² cos u¹)
(r cos u ²  cos  u¹        -r sin u ² sin u¹)

(0 .......................r cos u ²)

 

Example 2: A cone of revolution with apex x= (0, 0 ,0) and with x ₃ - axis as axis of revolution i.e.

Can be represented in the form:

a ( x ₁² +   x ₂ ²) -  x ₃²    = 0  

And:   x ₃    = + a Ö(x ₁ ² +   x ₂ ²)

represents one of two graphics, x ₃  >  o   and  x ₃    <  0 of the cone depending on the sign of the square root.  In the case shown then we have;  x ₃  = aÖ(x ₁ ² +   x ₂ ²).  The parametric representation can be written:

x ( u¹,  u ² )  =  ( u¹  cos u², u¹ sin u ² , au ²) 

Note the curves u¹  = const. are curves parallel to the x ₁ x ₂ - plane while the curves u¹  = const. are the generating straight lines of the cone.

Suggested Problems:

1) For the given cone of revolution (Example 2) write the corresponding Jacobian (matrix).

2) Find a parametric representation for each of the following:

a) Ellipsoid:  x ( u¹,  u ² )  =   (a cos u² cos u¹ , b cos u² sin u¹, c sin u ²)

b)    Hyperbolic paraboloid:  

x ( u¹ ,  u ²)  =  (au¹ cosh u²,  bu¹ sinh u²,  (u¹ ) ²)